How did they film Jim Craig's descent down that steep hill?

Actually both answers are partially wrong and partially right. Burlinson definitely did not perform all of his own stunts; however, he has confirmed that the scene of him riding down the cliff was done by him. But it looks like it was also done by a stunt man, named Gerald Egan.

From wikipedia:

Tom Burlinson has confirmed that it was definitely him who rode the horse down the side of the mountain for the "terrible descent" during the dangerous ride—commenting that he had been asked about this numerous times, and that he became known as "The Man from Snowy River" because of his ride. Burlinson had ridden a horse only a few times before being cast in the film. He was taught to ride by mountain cattleman Charlie Lovick, who owned the buckskin horse Burlinson rode in the film. Gerald Egan doubled for Burlinson for several riding shots in the film, including the jump into the "terrible descent". Other moments in the film such as when Jim is thrown over the fence into the path of the brumbies were performed by professional stunt men. Nevertheless, Burlinson did much more of the action riding in the film than an actor normally would, including all the profile shots of the downhill ride.

Tom Burlinson, who played Jim Craig, performed all his own horse-riding stunts in The Man from Snowy River. He was not an experienced horse rider before making this movie. When he took Denny over the cliff to go after the brumbies, it was a one-take shot at full gallop down the cliff face.

It's the Aussie way! See this site.

Unfortunately, this is wrong. My wife (who had a brief role in the second film) is from Mansfield, Vic. (the nearest town) and is certain that a local (Gerald Egan) performed the famous downhill ride. A lot of well-known local horsemen were included in the film, uncredited.

You are quite correct above. Tom did NOT do this stunt. Gerald did. I was there when they made that movie, my hubby at the time was a mate of Gerald's and Gerald did it. Gerald spent the day walking up and down the hill with the horse, letting the horse know and sense every step, then did the one time take down. I get quite angry when people say Tom did it, when he didn't. Yes Tom did do some stunts but not the dangerous ones.